Currents in matrix elements

In summary, there are two types of currents in the expression for a matrix element for an interaction - scalar and vector. The scalar current is formed by multiplying an adjoint spinor and a spinor, while the vector current is formed by multiplying an adjoint spinor, a gamma matrix, and a spinor. The confusion arises when trying to understand the vector current, as it involves contracting three indices - the vector index, a spinor index, and an adjoint spinor index. This results in a vector quantity, despite the initial expectation of a scalar or another spinor. Additionally, when dealing with a matrix element with multiple gamma matrices, the expression must be summed over all four terms, each with a different gamma matrix.
  • #1
alsey42147
22
0
i'm a bit confused about the currents in the expression for a matrix element for an interaction...

e.g. you could have a current like (adjoint spinor)x(spinor) which is scalar, this makes sense to me.

or you could have a current like (adjoint spinor)x(gamma matrix)x(spinor) which is vector according to all the books I've looked at. i don't get this - i would have thought that (gamma matrix)x(spinor) is either a vector or another spinor or something with 4 components, but then multiplying that by the adjoint spinor would just leave you with a scalar with 1 component. I'm guessing this is wrong but i can't see why.

also, say the matrix element looks something like:

(number)x[(adjoint)x(gamma-mu)x(spinor)]x[(adjoint)x(gamma-mu)x(spinor)]

with one gamma-mu having the index up, and the other one index down; does this mean that i sum the above expression over mu; i.e. a sum of 4 terms each with a different gamma matrix?

thanks in advance, and apologies for my lack of latex skills.
 
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  • #2
remember that the gamma matrices actually have THREE indices! Writing it out explicity, they are [itex]\gamma^\mu_{\dot{\alpha}\beta}[/itex]. The [itex]\mu[/itex] index is the vector index, the undotted lower index is a spinor index and the dotted lower index is an "adjoint spinor" index. So you must contract ALL of these indices together:

[tex]\bar{\psi}\gamma^\mu\psi\equiv \bar{\psi}^{\dot{\alpha}}\gamma^\mu_{\dot{\alpha}\beta}\psi^{\beta}[/tex]

and so it is a vector. Without the adjoint spinor, it would be this hybrid object with two indices.
 
  • #3


Thank you for your question. The concept of currents in matrix elements can be a bit confusing, but let me try to clarify it for you.

First, let's start with the definition of a current. In quantum field theory, a current is a conserved quantity that describes the flow of some physical quantity. In the context of matrix elements, a current represents the flow of some quantity between the initial and final states of an interaction.

Now, let's look at the two examples you mentioned. In the first example, the current is (adjoint spinor)x(spinor). This represents the flow of a scalar quantity between the initial and final states. This makes sense because the product of an adjoint spinor and a spinor is a scalar.

In the second example, the current is (adjoint spinor)x(gamma matrix)x(spinor). This represents the flow of a vector quantity between the initial and final states. This may seem confusing because, as you mentioned, the product of a gamma matrix and a spinor can be a scalar, vector, or another spinor. However, in this case, the adjoint spinor ensures that the product is a vector. The adjoint spinor acts like a "vector index" that transforms the product of a gamma matrix and a spinor into a vector.

Now, let's consider the matrix element you mentioned:

(number)x[(adjoint)x(gamma-mu)x(spinor)]x[(adjoint)x(gamma-mu)x(spinor)]

This expression represents the matrix element for an interaction that involves the exchange of a vector particle (represented by the gamma-mu term). The number represents the strength of the interaction, while the adjoint spinors and gamma matrices represent the initial and final states of the particles involved in the interaction.

To answer your question about the summation over mu, yes, you would sum over all four terms with different gamma-mu matrices. This is because the matrix element represents the contribution from all possible interactions involving the exchange of a vector particle in different directions.

I hope this helps clarify the concept of currents in matrix elements for you. If you have any further questions, please don't hesitate to ask.
 

1. What are matrix elements?

Matrix elements are numerical values that represent the relationship between two physical quantities in a matrix representation. They are used in quantum mechanics to describe the transition between two quantum states.

2. How are matrix elements calculated?

Matrix elements are calculated by taking the inner product of two quantum states, which involves taking the complex conjugate of one state and multiplying it by the other.

3. What is the significance of currents in matrix elements?

Currents in matrix elements represent the flow of a physical quantity between two quantum states. They are used to calculate the probability of a transition occurring between two states.

4. How do matrix elements relate to quantum mechanics?

Matrix elements play a crucial role in quantum mechanics as they are used to describe the behavior of quantum systems and calculate the probability of different outcomes. They are also used in the theoretical framework of quantum field theory.

5. Can matrix elements be measured experimentally?

Yes, matrix elements can be measured experimentally through techniques such as spectroscopy or scattering experiments. These measurements can provide valuable insights into the behavior of quantum systems and validate theoretical predictions.

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